Arithmetic logic unit and method for numerical computations in Galois fields

ABSTRACT

An integrated circuit for error correction takes advantage of a novel data representation (&#34;tower representation&#34;) for a selected finite Galois field. Using this representation, novel circuits which utilize the hierarchical structures in the subfields of the selected finite Galois field can be constructed. In one embodiment, GF(256) multipliers, GF(256) multiplicative inverse circuits, GF(256) logarithm circuits can be constructed out of GF(16) multipliers, GF(16) multiplicative inverse circuits and other GF(16) components. These GF(16) components, in turn, can be constructed from still simpler GF(4) components. In that embodiment, a user-programmable burstlimiter is provided. In that embodiment also, a novel quadratic equation solver is provided.

CROSS REFERENCE TO A RELATED PATENT APPLICATION

This application is a division of U.S. patent application Ser. No.08/542,262, filed Oct. 12, 1995, of the same assignee now U.S. Pat. No.5,812,438.

BACKGROUND OF THE INVENTION

1. Field of the Invention

This invention relates to electronic circuits for ensuring dataintegrity in data storage or data communication applications; inparticular, this invention relates to electronic circuits for numericalcomputation in Galois fields useful in such applications.

2. Discussion of the Related Art

Error correction and error detection codes have been used extensively indata communication and data storage applications. In a datacommunication application, data is encoded prior to transmission, anddecoded at the receiver. In a data storage application, data is encodedwhen stored in a storage device, e.g. a disk drive, and decoded whenretrieved from the storage device. For the present discussion, it isunnecessary to distinguish between these applications. Hence, althoughthe remainder of this description describes a data storage-retrievalsystem, the principles discussed herein are equally applicable to a datacommunication application.

In a typical application of error detection and correction codes, datasymbols are stored in blocks. Each such block includes a selected numberof special symbols, called check symbols. A symbol may consist of asingle bit or multiple bits. The check symbols in each block representredundant information concerning the data stored in the block. Whendecoding the blocked data, the check symbols are used to detect both thepresence and the locations of errors and, in some instances, correctthese errors. The theory and applications of error correction codes aredescribed extensively in the literature. For example, the texts (i)"Error-Correcting Codes", Second Edition, by W. Wesley Peterson and E.J. Weldon, published by the MIT Press, Cambridge, Mass. (1972), and (ii)"Practical Error Correction Design for Engineers", revised secondedition, by N. Glover and T. Dudley, Cirrus Logic, Colorado, publisher(1991), are well-known to those skilled in the art.

In a typical application of error correction codes, the input data isdivided into fixed-length blocks ("code words"). Each code word consistsof n symbols, of which a fixed number k are data symbols, and theremaining (n-k) symbols are check symbols. (For convenience, in thisdescription, such a code is referred to as an (n, k) code). As mentionedabove, the check symbols represent redundant information about the codeword and can be used to provide error correction and detectioncapabilities. Conceptually, each data or check symbol of such a codeword represents a coefficient of a polynomial of order (n-1). In theerror correcting and detecting codes of this application, the checksymbols are the coefficients of the remainder polynomial generated bydividing the order (n-1) polynomial by an order (n-k) "generator"polynomial over a Galois field. For a discussion of Galois fields, thereader is directed to §6.5 in the aforementioned text "Error-CorrectingCodes" by W. Peterson and E. Weldon Jr. For an order (n-1) polynomialdivided by an order (n-k) polynomial, the remainder polynomial is oforder (n-k-1). Typically, in a data storage application, both the datasymbols and the check symbols are stored.

During decoding, both data symbols and check symbols are read from thestorage medium, and one or more "syndromes" are computed from the codeword (i.e. the data and the check symbols) retrieved. A syndrome is acharacteristic value computed from a remainder polynomial, which isobtained by dividing the code word retrieved by the generatorpolynomial. Ideally, if no error is encountered during the decodingprocess, all computed syndromes are zero. In some applications, e.g. incertain cyclic redundancy check schemes, a non-zero characteristicnumber results when no error is encountered. Without loss of generality,a syndrome of zero is assumed when no detectable error is encountered. Anon-zero syndrome indicates that one or more errors exist in the codeword. Depending on the nature of the generator polynomial and the typeof error to be detected and corrected, the encountered error may or maynot be correctable.

A well-known class of error correcting codes is the Reed-Solomon codes,which are characterized by the generator polynomial G(X), given by:

    G(X)=(X+α.sup.j) (X+α.sup.j+1) (X+α.sup.j+2) . . . (X+α.sup.j+i-1) (X+α.sup.j+i)

where α is a basis element of GF(2^(m)) and, i and j are integers.

Because errors often occur in bursts, a technique, called"interleaving", is often used to spread the consecutive error bits orsymbols into different "interleaves", which can each be correctedindividually. Interleaving is achieved by creating a code word of lengthnw from w code words of length n. In one method for forming the new codeword, the first w symbols of the new code word are provided by the firstsymbols of the w code words taken in a predetermined order. In the samepredetermined order, the next symbol in each of the w code words isselected to be the next symbol in the new code word. This process isrepeated until the last symbol of each of the w code words is selectedin the predetermined order into the new code word. Another method tocreate a w-way interleaved code is to replace a generator polynomialG(X) of an (n, k) code by the generator polynomial G(X^(W)). Thistechnique is applicable, for example to the Reed-Solomon codes mentionedabove. Using this new generator polynomial G(X^(W)), the resulting (nw,kw) code has the error correcting and detecting capability of theoriginal (n, k) code in each of the w interleaves so formed.

An error detection and correction system requires extensive calculationin finite Galois fields. In the prior art, such calculation is carriedout by essentially unstructured customized random logic circuitsconsisting of logic gates and linear feedback shift registers. Theselogic circuits are complicated and relatively slow.

SUMMARY OF THE INVENTION

In accordance with the present invention, an error correcting codeintegrated circuit using a novel data representation for calculatingerror detecting and correcting codes and a method for providing such anintegrated circuit are provided. The novel data representation includesa hierarchical data structure such that a value in a GF(q²) finiteGalois field, can be represented by two values in a GF(q) finite Galoisfield. Consequently, logic circuits for arithmetic and logic operationsin GF(q²) can be constructed out of logic circuits for logic circuitsfor arithmetic and logic operations in GF(q). According to the presentinvention, novel logic circuits for multiplication, calculatingmultiplicative inverses, and calculating logarithms are provided,exploiting the hierarchical structure of the novel data representation.

In accordance with another aspect of the present invention, a quadraticequation solver circuit is provided. Using this quadratic equationsolver, uncorrectable errors are detected by testing the mostsignificant bit of a characteristic value of the quadratic equation.

In accordance of another aspect of the present invention, in aninterleaved implementation, additional "overall" syndromes are providedby summing (i.e. bit-wise exclusive-OR) corresponding syndromes of allthe interleaves.

In accordance with an other aspect of the present invention, aburstlimiting circuit is provided which utilizes a user-programmableburstlength.

The present invention is better understood upon consideration of thedetailed description below and the accompanying drawings.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of an integrated circuit 100, which is anembodiment of the present invention.

FIG. 2 is a block diagram of encoder/decoder 101 in integrated circuit100 of FIG. 1.

FIGS. 3a-3d provides, for this embodiment, the logic circuitsimplementing encoders 201a-201d, respectively.

FIG. 4 shows a logic circuit of alpha module 201a for decoding.

FIG. 5 shows a logic circuit for implementing register 206a.

FIG. 6 is a block diagram showing the functional blocks of errorevaluation and location block 104.

FIG. 7 shows a logic circuit which implements holding registers 105 inthe embodiment of the present invention shown in FIG. 1.

FIG. 8 is a block diagram of error value and location calculator 603.

FIG. 9 is a block diagram of arithmetic logic unit 801.

FIG. 10 is a block diagram of GF'(256) multiplier 903 in arithmeticlogic unit 801, provided in accordance with the present invention.

FIG. 11 shows a block diagram of GF'(16) multiplier 1005 of FIG. 10.

FIG. 12 shows auxiliary multiplier 1008 of GF'(256) multiplier 903 ofFIG. 10.

FIG. 13 shows a logic circuit for auxiliary multiplier 1112 in GF'(16)multiplier 1005 of FIG. 11.

FIG. 14 shows a logic circuit of GF'(4) multiplier 1105 in GF'(16)multiplier 1005 of FIG. 11.

FIG. 15 is a block diagram of GF'(256) multiplicative inverse circuit901, in accordance with the present invention.

FIG. 16 is a block diagram of GF'(16) multiplicative inverse circuit1512 in GF'(256) multiplicative inverse circuit 901 of FIG. 15, inaccordance with the present invention.

FIG. 17 is a logic circuit 1700 for GF'(4) multiplicative inversecircuit 1612 in GF'(16) multiplicative inverse circuit 1512 of FIG. 16,in accordance with the present invention.

FIG. 18 is a flow diagram illustrating a method for solving a quadraticequation.

FIG. 19 is a logic circuit implementing matrix multiplier 908 ofarithmetic and logic unit 801 of FIG. 8.

FIG. 20 is a block diagram of logarithm unit 607 of error evaluation andlocation unit 104.

FIG. 21 is a flow diagram of the operation of error evaluation andlocation block 104.

FIG. 22 is a flow diagram showing a process for (a) converting errorlocations within each interleave to error locations relative to thebeginning of the check symbol field in a 1024-byte sector; and (b)sorting the error locations in ascending order.

FIG. 23a and 23b together form a flow diagram showing a process for theburstlimiting and error correction operations of burstlimiter 103.

FIG. 24 shows byte positions and bit positions as numbered in thedescriptions of the processes shown in FIGS. 23a and 23b.

DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS

The present invention achieves computation in finite Galois fields usinga full-quadratic tower representation approach ("tower representation").As will be shown below, the circuits of the present invention under thetower representation are highly modular and carry out themultiplication, division, logarithm and solution of quadratic equationoperations efficiently. Consequently, design of logic circuits in largefinite Galois fields is thereby greatly simplified. Performance of suchcircuits are also made efficient thereby.

The present invention is applicable to Reed-Solomon codes represented bya number of bits which is a power of 2, e.g. 2, 4, 8 . . . , 2^(k) bits,for k≧1. In the embodiment described below, 8-bit Reed-Solomon codes areused. The present invention can be, of course, extended to anyReed-Solomon Codes represented by 2^(k) bits, k≧1. In this description,a Galois field represented by m bits per element is denoted GF(2^(m)).Thus, the present invention is applicable to Galois fields GF(2), GF(4),GF(16), GF(256), GF(65536), . . . GF(2².spsp.k), for k≧0.

In the embodiment described below, the following Galois fields are usedto illustrate the present invention:

(i) GF'(2)={0,1};

(ii) GF'(4)={0,1,β,β+1}, where β is a basis element of the finite Galoisfield GF'(4) in which β² +β+1=0;

(iii) GF'(16)={x.linevert split.x=aγ+b;a,bεGF'(4)}, where γ is a basiselement of the finite Galois field GF'(16) in which γ² +γ+β=0; and

(iv) GF'(256)={x.linevert split.x=cδ+d;c,dεGF'(16)}, where δ is a basiselement of the finite Galois field GF'(256) in which δ² +δ+ε=0, andε=βγ+γ;

In other words, using Galois fields defined in (i)-(iv) above, a GF'(4)value can be represented by two GF'(2) values, a GF'(16) value can berepresented by two GF'(4) values, and a GF'(256) value can berepresented by two GF'(16) values. Of course, this method ofrepresenting values in a higher order Galois field by two values of alower Galois field can be extended ad infinitum.

FIG. 1 is a block diagram of an integrated circuit 100, which is anembodiment of the present invention. As shown in FIG. 1, integratedcircuit 100 includes an encoder/decoder block 101, holding registers105, control circuit 102, error evaluation and location block 104, andburstlimiter 103. Encoder/decoder block 101 includes an encoder which,during encoding, receives 8-bit data symbols on bus 110 to generateReed-Solomon code words on buses 111a-111d. Encoder/decoder block 101also includes a decoder which, during decoding, receives Reed-Solomoncode words on bus 110 and provides both the decoded 8-bit data symbolsand the 8-bit syndromes of the Reed-Solomon code words received. Asexplained above, the syndromes thus calculated are used for errordetection and correction.

During decoding, after the final syndromes are computed and receivedinto holding registers 105, circuits in error evaluation and locationblock 104 detects from the computed syndromes whether one or more errorshave occurred and computes, when possible, the values and locations ofsuch errors. In the present embodiment, to reduce the probability ofmiscorrection, not all detected errors are corrected. Burstlimiter 103,as described below, detects the occurrence of two burst errors andlimits correction of errors to bursts of a predetermined length. Thepredetermined length is a programmable value, in accordance with thepresent invention.

For a three-interleve implementation, integrated circuit 100 of thepresent embodiment uses a generator polynomial of the form:

    GP(X)=(X.sup.3 +1) (X.sup.3 +α) (X.sup.3 +α.sup.2) (X.sup.3 +α.sup.3)

where α is a basis element of the finite Galois field over which thegenerator polynomial operates.

Generator polynomials and their used for generating Reed-Solomon codesare discussed in detail in copending patent application ("CopendingPatent Application"), entitled "Global Parity Symbol for InterleavedReed-Solomon Coded Data", by Frank S. Lee et al, Ser. No. 08/270,858,filed on Jul. 5, 1994, assigned to Adaptec, Incorporated, which is alsothe assignee of the present application. The disclosure of the CopendingPatent Application is hereby incorporated by reference in its entirety.The generator polynomial of the present embodiment provides 512 datasymbols and 12 check symbols. In the present embodiment, each sector is1024 bytes. Therefore, in each 1024-byte sector, the present embodimentstores 259 undefined bytes, which is followed by 241 bytes of leadingzeroes. The leading zeroes are, in turn, followed by 512 data symbolsand 12 check symbols.

Encoder/decoder 101 is shown in further detail in FIG. 2. As shown inFIG. 2, encoder circuits 201a-201d each implement one factor of thegenerator polynomial GP shown above. During encoding, input data arriveon 8-bit data bus 110 and are selected by multiplexer 213 onto 8-bitdata bus 212. Multiplexer 213 selects between data on data bus 110 andtest bus 115a. Test bus 115a is provided for testing encoder/decodercircuit 101, when a test control signal on test bus 116 is asserted.During encoding, however, each 8-bit value of data bus 212 is the inputdata received on data bus 110, and multiplexers 205a-205d select theoutput data of the 8-bit values of busses 209a-209d, respectively. The8-bit values of busses 209a-209d are, respectively, the output values ofencoder 201a, and summers 203a-203c. The output values of summers203a-203c are the sums (i.e. bitwise exclusive-ORs) of the 8-bit valueon data bus 212 and each of the output 8-bit values of encoders201b-201c, respectively. Registers 206a-206d each include three cascaded8-bit component registers, thereby performing the task of delaying bythree clock periods the output of the 8-bit value each registerreceives. Propagation of 8-bit values through each of registers206a-206d is controlled by three clock signals on 3-bit bus 117. Eachcomponent register inside each of registers 206a-206d can beindividually reset by a reset signal on 3-bit control bus 118.

Encoder 201a-201d each receive from data bus 215, for every clocktransition on any of the three clock signals of 3-bit bus 117, the sumof the data on bus 212 and an output 8-bit value of register 206d.Encoder 201a-201d provide output 8-bit values on bus 209a and busses213a-213c respectively, except when outputting check symbols, when the8-bit input values to encoders 201a-201d on bus 215 are set to zero bythe inactive control signal 214 ("datafield"). The output data symbolsappear on 8-bit bus 211d, at every clock transition of the three clocksignals of 3-bit bus 117. Each clock signal on 3-bit bus 117 correspondsto a data interleave. Multiplexers 207a-207d select the output values onbusses 211a-211d respectively.

FIGS. 3a-3d provides, for this embodiment, the logic circuitsimplementing encoders 201a-201d, respectively. FIG. 5 shows a logiccircuit for implementing register 206a. The logic circuits forimplementing registers 206b-206d are each identical to the logic circuitof FIG. 5 for register 206a.

Returning to FIG. 2, during decoding, multiplexers 205a-205d each selectthe 8-bit values on busses 210a-210d, respectively. The 8-bit value onbus 210a is the sum of the output value of register 206a and the input8-bit encoded data symbol on bus 212. The 8-bit values on busses210b-210d are the sums of the input encoded data symbol and acorresponding one of the 8-bit output values of alpha modules 202a-202con busses 216a-216c respectively. Each of alpha modules 202a-202cmultiplies its input 8-bit value, which is the output value of acorresponding register of registers 206a-206c, to a corresponding one ofthe values (α,α², α³ provided in the generator polynomial GP(x) shownabove. FIG. 4 shows a logic circuit 400 used to implement alpha module202a. The logic circuits of alpha modules 202b and 202c are formed bycascading logic circuit 400 for alpha module 201a two and three timesrespectively.

Holding registers 105 include twelve 8-bit registers, and six 2-bitregisters, which can be combined with six of the twelve 8-bit registersto form 10-bit registers for holding 10-bit values for error location.The logic circuit for implementing holding registers 105 is provided inFIG. 7. As shown in FIG. 7, the twelve 8-bit registers are 711a, 711b,711c, 711d, 714a, 714b, 714c, 714d, 716a, 716b, 716c, and 716d. Inaddition, 2-bit registers 711b1, 711d1, 714b1, 714d1, 716b1 and 716d1can be combined with 8-bit registers 711b, 711d, 714b, 714d, 716b and716d to form six 10-bit registers for error location. Holding registers105 receives 8-bit input values on busses 111a-111d from encoder/decoderblock 101, 8-bit values on data busses 120-122 from error evaluation andlocation block 104 and 8-bit values on data busses 131, 138, 139, 702,and 703 from burstlimiter 103. Holding registers 105 also receives 2-bitvalues on data busses 702a and 703a from burstlimiter 103. The 2-bitvalues on data busses 702a and 703a are combined with the 8-bit valueson data busses 702 and 703 to form 10-bit values for error locationdata. 6-bit control busses 132-137 are provided by burstlimiter 103 tocontrol multiplexers 710a-710d, 712a-712d, and 715a-715d to select fromthe input data on the data busses at the input terminals of multiplexers701a-701d, 712a-712d and 715a-715d to designated registers of holdingregisters 105. In addition to receiving input values during encoding anddecoding, holding registers 105 are also used as sources of operands anddestination of results in calculations performed by error evaluation andlocation block 104. For decoding, holding registers 105 are used toprovide six error location registers and six error value registers.

As mentioned above, error evaluation and location block 104 calculateserror location and error values after the final four syndromes arecomputed. FIG. 6 is a block diagram showing the functional blocks oferror evaluation and location block 104. As shown in FIG. 6, errorevaluation and location block 104 receives six 8-bit data values fromdata busses 140-145, corresponding to error data values stored inregisters 716a, 716c, 714a, 714c, 711a and 711c of holding registers105. Error evaluation and location block 104 also receives 8-bit datafrom data busses 146-151, corresponding to error location values storedin registers 711d, 711b, 714d, 714b, 716d and 716b of holding registers105. In addition, error evaluation and location block 104 also receives8-bit data from test data busses 115a-115d. 4-to-1 Multiplexers601a-601d, multiplexes 8-bit data from all data busses 115a-115d anddata busses 151 onto data busses 602a-606d. As shown in FIG. 6, testdata busses 115a-115d can be multiplexed onto data busses 602a-602drespectively. Error data values on data busses 141-145 can be eachmultiplexed onto a designated one of data busses 602a and 602c.Likewise, error location values on data busses 146-151 can be eachmultiplexed onto a designated one of data busses 602b and 602d. Datavalues on data busses 602a-602d are provided to an error value andlocation calculator 603, which calculates the error values and locationsunder the control of routines stored in a 32×14 bytes read-only memory(ROM) 606. Each routine in ROM 606 is a sequence of instructions decodedby instruction decode unit 604. The output values of error value andlocation calculator 603 can also affect the sequence of instructionexecutions in a routine of ROM 606 by conditional branching. Conditionalbranches are effectuated by a branch check unit 605, which tests thevalues on 8-bit busses 608a and 609 at the output terminals of errorvalue and location calculator 603. Branch check unit 605 providescontrol signals "BRANCH" (branch) and "NBRANCH" (no branch) at terminals610 and 611. ROM 606, branch check unit 605, and instruction decode unit604 form a state machine to control the operations of error evaluationand location block 104.

Four types of instructions are provided from ROM 606: a) branchinstructions, b) arithmetic and logic operations (including aninstruction to take the logarithm of an 8-bit value), c) loadinstructions and d) an instruction to solve a quadratic equation.

Error evaluation and location block 104 also includes a logarithm unit607 which takes the logarithmic value of an 8-bit value on data bus 608aoutput from error value and location calculator 603.

FIG. 8 is a block diagram of error value and location calculator 603. Asshown in FIG. 8, multiplexers 802a, 802b and 802c multiplex input 8-bitdata from data busses 602a-602d, 608a, 609, 807 and 810 onto data busses806a-806b. The data busses 608a, 609, and 807 provide output data fedback from the output data of arithmetic logic unit 801. 8-bit values ondata busses 806a-806b are provided as input values to arithmetic logicunit 801. The output values of arithmetic logic unit 801 are provided on8-bit data bus 122. The output value of arithmetic logic unit 801 isalso received by each of registers 803a, 803b and 804. Multiplexers805a-805c multiplex the 8-bit output values of registers 803a, 803b and804 onto data busses 608a, 608b and 807.

FIG. 9 is a block diagram of arithmetic logic unit 801. As shown in FIG.9, arithmetic logic unit 801 includes a multiplicative inverse circuit901, a multiplier 903, an adder 906, and a matrix multiplier 908.Arithmetic logic unit 801 operates over the finite Galois fieldGF'(256). The operands for arithmetic logic unit 801 are taken from the8-bit values on busses 806a-806c. Multiplicative inverse circuit 901receives an 8-bit input datum on data bus 806a. Multiplier 903 receivesoperands from data busses 910 and 806a. Multiplexers 902 selects betweenthe 8-bit output value of multiplicative inverse circuit 901 and the8-bit value on data bus 806a. The result value of multiplier 903 isprovided on data bus 911. Multiplexer 904 selects between the 8-bitvalue on data bus 806b and the 8-bit output value of multiplier 903. The8-bit output value of multiplexer 904 is provided on data bus 912. The8-bit value on data bus 912 and the 8-bit value on data bus 806c areprovided as operands to summer 906. Matrix multiplier 908 receives an8-bit input value on data bus 806b and provides an 8-bit output value ondata bus 914. Matrix multiplier 908 is used in a step for solving thequadratic equation y² +ay+b=0. The value provided on data bus 806b isthe value ##EQU1## and the output value of matrix multiplier 908 is avalue x, which is related to the roots y₁ and y₂ of equation y² +ay+b=0by the equations: y₁ =ax; y₂ =ax+a . Multiplexer 907 selects onto 8-bitoutput bus 122 either the 8-bit output datum of matrix multiplier 908 ondata bus 914 and the 8-bit output datum of summer 906 on bus 913.

Under the present invention multiplication in GF'(256) can be performedby taking advantage of the hierarchical structure of these Galoisfields. Multiplication in GF'(256) can be performed in GF'(16) withGF'(16) multipliers, which perform the multiplication in GF'(4) usingGF'(4) arithmetic and logic operations. This approach can be, of course,easily extended to GF'(65536) to take advantage of GF'(256) multipliers.Clearly, the technique also can be extended to higher Galois fields.

In a multiplication involving two GF'(4) values P and R, where P can berepresented by (pβ+q) and R can be represented by (rβ+s),p, q, r,sεGF'(2), the following relation (1) holds:

    P*R=(pβ+q) (rβ+s)=((p+q) (r+s)+qs)β+qs+pr

Thus, it can be seen that a multiplication in GF'(4) can be carried outby arithmetic operations (i.e. multiplication and addition) in GF'(2).Likewise, a multiplication in GF'(16), involving the values E and G,expressed as (eγ+f) and (gγ+h) respectively, where e, f, g, hεGF'(4),can be calculated using GF'(4) arithmetic operations according to therelation (2):

    E*G=(eγ+f) (gγ+h)=((e+f) (g+h)+fh)γ+fh+egβ

Furthermore, multiplying a GF'(4) value (fγ+h) with β can beaccomplished in GF'(4) arithmetic using the relation (3):

    (fβ+h)β=(f+h)β+f

Similarly, a multiplication in GF'(256) can be performed in GF'(16) byrepresenting the GF'(256) operands A and C respectively as (aδ+b) and(cδ+d), a, b, c, dεGF'(16), and using the relation (4):

    A*C=(aδ+b) (cδ+d)=((a+b) (c+d)+bd)δ+bd+acε

The multiplication by ε can also be performed in GF'(16) using therelation (5):

    ((rβ+s)γ+(tβ+u))ε=((s+u)β+(r+s+t+u))γ+r.beta.+s

where (s+u)β+(r+s+t+u) and rβ+s are in GF'(4) and r, s, t, uεGF'(2).Multiplication by β can, of course be performed in GF'(4), in accordancewith the relation (3) already provided above.

In accordance to the present invention, GF'(256) multiplier 903 isprovided by the logic circuit which block diagram is shown in FIG. 10.As shown in FIG. 10, an 8-bit input operand on bus 910 is separated intotwo 4-bit operands on 4-bit busses 1001a and 1001b, in accordance withthe present invention. The higher order 4 bits, i.e. the value on bus1001a, and the lower order 4 bits, i.e. the value on bus 1001b, form thetwo GF'(16) values for representing the GF'(256) value on bus 910. (Thebusses on FIG. 10 are annotated with the variables a, b, c, and d ofrelation (4) to assist the reader in following the data flow inmultiplier 903). Likewise, the 8-bit operand on bus 806b is split intotwo 4-bit operands (i.e. GF'(16)) on busses 1002a and 1002b. The 4-bitoperands on busses 1001a and 1001b are summed (i.e. bitwiseexclusive-ORed) in summer 1003 to provide a 4-bit value on bus 1012, andthe 4-bit operands on busses 1002a and 1002b are similarly summed insummer 1004 to provide a 4-bit value on bus 1013. In addition, the 4-bitoperand on bus 1001b is multiplied to the 4-bit operand on bus 1002b inGF'(16) multiplier 1005 to provide a 4-bit value on bus 1011. Similarly,GF'(16) multiplier 1006 multiplies the 4-bit operands on busses 1001aand 1002a to provide a 4-bit value on bus 1014.

The 4-bit results of summers 1003 and 1004 on busses 1012 and 1013 aremultiplied in GF'(16) multiplier 1007 to provide a 4-bit value on bus1016. This 4-value on bus 1016 and the 4-bit result from GF'(16)multiplier 1015 are summed in summer 1009 to provide a 4-bit result onbus 911a, providing thereby the higher 4 bits of bus 911. The 4-bitresult of GF'(16) multiplier 1006 on bus 1014 is"auxiliarily-multiplied" in auxiliary multiplier 1008 to provide a 4-bitresult on bus 1015, which is summed in summer 1010 to the 4-bit resultof GF'(16) multiplier 1005 on bus 1011 to provide a 4-bit value on bus911b. Any element of GF'(16) having the bit value `1` as itsmost-significant bit can be chosen as auxiliary multiplier 1008. Ingeneral, in going from GF'(q) to GF'(q²), any element of GF (q) having abit value of `1` in its most significant bit can be chosen as anauxiliary multiplier. The 4-bit result on bus 911b forms the lower 4bits of bus 911. The 4-bit values on bus 911a and 911b, which are thetwo GF'(16) components of a GF'(256) value, form the 8-bit result ofGF'(256) multiplier 911.

A logic circuit for auxiliary multiplier 1008 is shown in FIG. 12. Asshown in FIG. 12, the 4-bit value on bus 1014 is split into four 1-bit(i.e. GF'(2)) values at terminals 1201-1204. The notations r, s, t and uof relation (5) used in the above description are annotated at theterminals shown in FIG. 12 to assist the reader to follow the data flow.The 1-bit values at terminals 1201 and 1204 are summed by exclusive-ORgate 1205 to provide a value at terminal 1209. Likewise, the 1-bitvalues at terminals 1202 and 1203 are summed by exclusive-OR gate 1206to provide a 1-bit value at terminal 1208. The 1-bit values at terminals1208 and 1209 are summed at exclusive-OR gate 1207 to provide a 1-bitvalue at terminal 1210. The 1-bit values at terminal 1203 and 1204 arebuffered to provide additional signal strength at terminals 1211 and1212. The 1-bit values at terminals 1209, 1210, 1211 and 1212 form theGF'(16) result on bus 1015.

A GF'(16) multiplier can be implemented using logic circuits andmultipliers that operate in GF'(4). FIG. 11 shows a block diagram ofGF'(16) multiplier 1005 of FIG. 10. The logic circuit for GF'(16)multiplier 1005 also can be used to implement each of GF'(16)multipliers 1006 and 1007.

In accordance to the present invention, GF'(16) multiplier 1005 isprovided by the logic circuit which block diagram is shown in FIG. 11.As shown in FIG. 11, a 4-bit input operand on bus 1001b is separatedinto two 2-bit operands on 2-bit busses 1101a and 1101b, in accordancewith the present invention. The higher order 2 bits, i.e. the value onbus 1101a, and the lower order 2 bits, i.e. the value on bus 1101b, formthe two GF'(4) values for representing the GF'(16) value on bus 1001b.(The busses on FIG. 11 are annotated with the variables e, f, g, and hof relation (5) to assist the reader in following the data flow).Likewise, the 4-bit operand on bus 1002b is split into two 2-bitoperands (i.e. GF'(4)) on busses 1102a and 1102b. The 2-bit operands onbusses 1101a and 1101b are summed (i.e. bitwise exclusive-ORed) insummer 1103 to provide a 2-bit value on bus 1107, and the 2-bit operandson busses 1102a and 1102b are similarly summed in summer 1104 to providea 2-bit value on bus 1108. In addition, the 2-bit operand on bus 1101bis multiplied to the 2-bit operand on bus 1102b in GF'(4) multiplier1105 to provide a 2-bit value on bus 1109. Similarly, GF'(4) multiplier1106 multiplies the 2-bit operands on busses 1101a and 1102a to providea 2-bit value on bus 1110.

The 2-bit results of summers 1103 and 1104 on busses 1107 and 1108 aremultiplied in GF'(4) multiplier 1111 to provide a 2-bit value on bus1113. This 2-value on bus 1113 is then summed to the 2-bit result fromGF'(4) multiplier 1105 in summer 1115 to provide a 2-bit result on bus1011a, providing thereby the higher 2 bits of bus 1011. The 2-bit resultof GF'(4) multiplier 1106 on bus 1110 is "auxiliarily-multiplied" inauxiliary multiplier 1112 to provide a 2-bit result on bus 1114, whichis summed in summer 1116 with the 2-bit result of GF'(16) multiplier1105 on bus 1109 to provide a 2-bit value on bus 1011b. The 2-bit resulton bus 1011b forms the lower 2 bits of bus 1011. The 2-bit values on bus1011a and 1011b, which are the two GF'(4) components of a GF'(16) value,form the 4-bit result of GF'(4) multiplier 1005.

A logic circuit for auxiliary multiplier 1112 is shown in FIG. 13. Asshown in FIG. 13, the 2-bit value on bus 1110 is split into 2 1-bit(i.e. GF'(2)) values at terminals 1301 and 1302. The notations f and hof relation (3) used in the above description are annotated at theterminals 1301 and 1302 shown in FIG. 13 to assist the reader to followthe data flow. The 1-bit values at terminals 1301 and 1302 are summed byexclusive-OR gate 1303 to provide a 1-bit value at terminal 1304. The1-bit value at terminal 1302 is buffered to provide additional signalstrength at terminal 1305. The 1-bit values at terminals 1304 and 1305form the GF'(4) result on bus 1114.

FIG. 14 shows a logic circuit of GF'(4) multiplier 1105 in GF'(16)multiplier 1005 of FIG. 11, in accordance with multiplicative equationin GF'(2) disclosed above. Of course, other GF'(4) multipliers, such asGF'(4) multipliers 1106 and 1111 of FIG. 11, also can each beimplemented by the logic circuit of GF'(4) multiplier 1105.

GF'(256) multiplicative inverse circuit 901 can also be provided bytaking advantage of the hierarchical structure of the Galois fields ofthe present invention. Consider values A and G in GF'(256) representedby (aδ+b) and (gδ+h), where a, b, g, hεGF'(16). Since

    A*G=(aδ+b) (gδ+h)=(ah+bg+ag)δ+ag+bh

Thus, if (gδ+h) is the multiplicative inverse of (aδ+b), i.e. (aδ+b)⁻¹,then (ah+bg+ag)=0. Because a and b are independent, we obtain g=ka,h=k(a+b) for some kεGF'(16). Thus, the following relation (6) holds:

    (aδ+b).sup.-1 =(aδ+(a+b)) (b(a+b)+a.sup.2 ε).sup.-1

Similarly, for a value C in GF'(16), represented by (cγ+d), where c,dεGF(4), the following relation (7) holds:

    (cγ+d).sup.-1 =(cγ+(c+d)) (c.sup.2 β+d(c+d)).sup.-1

and for GF'(4), where e, fεGF'(2), the following relation (8) holds:

    (eβ+f).sup.-1 =(eβ+(f+e)) (e.sup.2 +f(e+f)).sup.-1

Using relations (6), (7) and (8), GF'(256) multiplicative inversecircuit 901 can be implemented by the circuit shown in block diagramform in FIG. 15. As shown in FIG. 15, multiplicative inverse circuit 901receives an 8-bit datum on bus 806a. The higher and lower 4-bits on bus806a, which form the two GF'(16) values to represent the 8-bit value onbus 806a, are provided on busses 1501a and 1501b. GF'(16) multiplier1502 squares the 4-bit value on bus 1501a to provide a 4-bit value onbus 1504, which is then auxiliarily-multiplied in auxiliary multiplier1506 to yield a 4-bit result on bus 1508. The logic circuits for anauxiliary multiplier and an GF'(16) multiplier are already describedabove with respect to auxiliary multiplier 1008 of FIG. 12 and GF'(16)multiplier 1005 of FIG. 11.

Concurrently, the 4-bit value on bus 1501a, i.e. the higher order 4-bitof bus 806a, is summed in summer 1503 with the 4-bit value on bus 1501b,i.e. the lower order 4-bit of bus 806a, to provide a 4-bit result on bus1505. The 4-bit result on bus 1505 is then multiplied in GF'(16)multiplier 1507 with the 4-bit input value on bus 1501b, to provide a4-bit value on bus 1509. The 4-bit values on busses 1508 and 1509 arethen summed in summer 1510, to provide a 4-bit value on 1511. The 4-bitmultiplicative inverse of the 4-bit value on bus 1511 is then found byGF'(16) multiplicative inverse circuit 1512. This 4-bit multiplicativeinverse is provided on bus 1513, which is then multiplied to the inputvalue on bus 1501a to provide an output value on 909a. Simultaneously,the 4-bit values on busses 1513 and 1505 are multiplied in GF'(16)multiplier 1515, to provide a 4-bit value on bus 909b. The two 4-bitGF'(16) values of busses 909a and 909b form the 8-bit GF'(256) valueoutput on bus 909. Again, to assist the reader to follow the data flow,the notations a and b, a, bεGF'(16), used above in relation (6), areprovided on the busses and terminals of GF'(256) multiplicative inversecircuit 901.

FIG. 16 is a block diagram of GF'(16) multiplicative inverse circuit1512 in GF'(256) multiplicative inverse circuit 901 of FIG. 15, inaccordance with the present invention. As shown in FIG. 16, GF'(16)multiplicative inverse circuit 1512 receives a 4-bit datum on bus 1511.The higher and lower two bits on bus 1511, which form the two GF'(4)values to represent the 4-bit value on bus 1511, are provided on busses1601a and 1601b. GF'(4) multiplier 1602 squares the 2-bit value on bus1501a to provide a 2-bit value on bus 1604, which is then auxiliarilymultiplied in auxiliary multiplier 1606 to yield a 2-bit result on bus1608. The internal structures of an auxiliary multiplier and a GF'(16)multiplier are already described above with respect to auxiliarymultiplier 1110 of FIG. 13 and GF'(4) multiplier 1105 of FIG. 14.

Concurrently, the 2-bit value on bus 1601a, i.e. the higher order twobits of bus 1511, is summed in summer 1603 with the 2-bit value on bus1601b, i.e. the lower order two bits of bus 1511, to provide a 2-bitresult on bus 1605. The 2-bit result on bus 1605 is then multiplied inGF'(4) multiplier 1607 with the 2-bit input value on bus 1501b, toprovide a 2-bit value on bus 1609. The 2-bit values on busses 1608 and1609 are then summed in summer 1610, to provide a 2-bit value on 1611.The 2-bit multiplicative inverse of the 2-bit value on bus 1611 is thenfound by GF'(4) multiplicative inverse circuit 1612. This 2-bitmultiplicative inverse is provided on bus 1613, which is then multipliedto the input value on bus 1601a to provide an output value on 1513a.Simultaneously, the 2-bit values on busses 1613 and 1605 are multipliedin GF'(4) multiplier 1615, to provide a 2-bit value on bus 1513b. Thetwo 2-bit GF'(4) values of busses 1513a and 1513b form the 4-bit GF'(16)value output on bus 1513. Again, to assist the reader to follow the dataflow, the notations c and d, c, dεGF'(4), used above in relation (7),are provided on the busses and terminals of GF'(16) multiplicativeinverse circuit 1512.

FIG. 17 is a logic circuit 1700 for GF'(4) multiplicative inversecircuit 1612 in GF'(16) multiplicative inverse circuit 1512 of FIG. 16.Logic circuit 1700 is-optimized by inspection of the four element Galoisfield GF'(4).

Included in arithmetic logic unit 801 is a matrix multiplier 908, whichis used in a step for solving the quadratic equation y² +ay+b=0. Asdescribed below, quadratic equations are solved to evaluate error valuesand error locations. It can be shown that the roots of the quadraticequation are ax₀ and ax₀ +a, for some x₀. x₀ can be found by a tablelook-up method, using the 8-bit value ##EQU2## Under the presentembodiment, using the Galois field GF'(256) of the tower representation,quadratic equation y² +ay+b=0 does not have roots in the Galois fieldGF'(256) for all values of c. In particular, if the leading bit of 8-bitvalue c is 1, no solution to the quadratic equation y² +ay+b=0 exists inGF'(256). To solve the quadratic equation y² +ay+b=0, the processillustrated by FIG. 18 is used. The steps illustrated in FIG. 18 can beaccomplished using the instructions in ROM 606, with the necessaryarithmetic or logic operations performed by arithmetic logic unit 801.As shown in FIG. 18, at decision point 1801, the value a is tested todetermine if it is a zero. If a is non-zero, then the error isdetermined to be uncorrectable, and procedure 1800 terminates at step1802. Otherwise, the value c is calculated at step 1803. (As discussedabove, in this embodiment, if the most significant bit of c is non-zero,a solution to quadratic equation y² +ay+b=0 does not exist in GF'(256)).Otherwise, the 8-bit value c is multiplied in step 1805 with aprecomputed an 8×8 matrix M, implemented by matrix multiplier 908, toprovide an 8-bit value x₀. In this multiplication, the 8-bit values X₀and c are treated as 1×8 vectors (i.e. matrix multiplier 908 implementsMC=X, where C and X are the 1×8 vectors, in which each element is a bitof the 8-bit value c or x₀). The roots y₁ and y₂ of quadratic equationy² +ay+b=0 are then computed in steps 1806 and 1807, using GF'(256)multiplier 903 and GF'(16) summer 906.

FIG. 19 is a logic circuit implementing matrix multiplier 908 ofarithmetic and logic unit 801 of FIG. 8.

Logarithm unit 607 (FIG. 6) provides a logarithm of a GF'(256). Themethod of logarithm unit 607 is also applicable to Galois fields underthe tower representation. Logarithm unit 607 takes advantage of thefollowing identity (9): ##EQU3## where (a,b)εGF'(q²); a, bεGF'(q), qbeing a power of 2, and α is a basis element of GF'(16). Instead ofstoring a value for each (a,b)εGF'(q²), only the values of log(a) and##EQU4## are stored. Thus, instead of a logarithm table for values inGF'(q²), which would normally require a memory with 2^(2q) locations,only 2^(q+1) locations are necessary. The logarithm of (a,b) is providedby summing the values log(a) and ##EQU5## in GF'(q²). A block diagram oflogarithm circuit 607 is shown in FIG. 20.

As shown in FIG. 20, logarithm circuit 607 receives an 8-bit GF'(256)value on bus 608b. The most significant four bits of bus 608b areprovided as 4-bit bus 2001a, and the remaining four bits are provided as4-bit bus 2001b. The two 4-bit values on busses 2001a and 2001b are theGF'(4) values representing the 8-bit GF'(256) value on bus 608a. Thevalue on bus 2001a is provided as an input value to a GF'(16)multiplicative inverse circuit 2002 to provide an 4-bit output value,which is then multiplied in GF'(16) multiplier 2004 to the 4-bit valueof bus 2001b to provide a 4-bit output value on bus 2012. Multiplexer2005 selects between the 4-bit values on bus 2001a and 2001b, accordingto the control signal at terminal 2003 output from NOR gate 2011. If the4-bit value on bus 2001a is zero, multiplexer 2005 outputs the 4-bitvalue of bus 2001b onto 4-bit bus 2013. Otherwise, multiplexer 2005outputs the 4-bit value of bus 2001b onto bus 2013. Both 4-bit values ofbusses 2012 and 2013 are provided to ROM 2006 to access their respectiveGF'(256) logarithmic values, provided on output busses 2008 and 2007,respectively. The 8-bit logarithmic values on busses 2007 and 2008 arethen summed (mod 256) in a GF'(256) adder (not shown) to provide thelogarithm value of the 4-bit value received on bus 608a. When the 4-bitvalue on bus 2001a is zero, multiplexer 2010 places a zero value on bus2008. In this embodiment, since the most significant four bits of eachlogarithmic value is the same as its least significant four bits, onlyfour bits of the logarithmic values are stored in ROM 2006.

To evaluate error values and locations, four syndromes S₀, S₁, S₂, andS₄ are computed per interleave. Thus, a total of twelve syndromes areprovided for each Reed-Solomon code word. FIG. 21 is a flow diagram ofthe operation of error evaluation and location block 104. As shown instep 2101 of FIG. 21, for each interleave, the values of syndrome S₀ andthe square of syndrome S₁ are tested if they are each zero. If both thevalues of syndrome S₀ and the square of syndrome S₁ are zero, the valuesof S₂ and S₃ are then checked at step 2103. If both S₂ and S₃ are thenfound to be zero, no error is found in the interleave and the syndromesof the next interleave are checked (i.e. returning to step 2101 via step2118). The process continues until all interleaves are checked. At step2102, if either the value S₀ or the value S₁ ² are non-zero, step 2104checks if both S₀ and S₁ ² are non-zero. If both S₀ and S₁ ² arenon-zero, at step 2105, the value ##EQU6## is computed. If either S₀ orS₁ ² is zero, then step 2106 is taken. At step 2105, the computed valueD is then used to check, at step 2107, if the conditions S₂ +S₁ *D=0 andS₃ +S₂ *D=0 are satisfied. If the conditions at step 2107 are satisfied,a single error is encountered. Error location is then provided by thevalue log(D), and the corresponding error value of this error is S₀.Otherwise, if the conditions at step 2107 are not satisfied, step 2106is taken.

At step 2106, a value "a" is computed according to the equation:##EQU7##

At step 2108, if the values of S₀ -S₃ satisfy either one of thefollowing equations, an uncorrectable error has occurred:

    S.sub.0 *S.sub.3 +S.sub.1 *S.sub.2 =0

    S.sub.0 *S.sub.1 +(S.sub.1).sup.2 =0

A value of "b" is then computed at step 2111 according to: ##EQU8## Atstep 2113, the value (S₂)² +S₁ *S₃ =0 is checked if it is equal to zero.If the value is zero, a value "c" defined by the following equation iscomputed at step 2114: ##EQU9##

At step 2115, the leading bit of the value c is then tested. If thisleading bit is non-zero, in accordance with our data representation, anuncorrectable error is detected. Otherwise, as discussed above, matrixmultiplier 908 is used to obtain the value of X₀, and the roots y₁ andy₂ of the quadratic equation y² +ay+b=0. At step 2117, the errorlocations corresponding to roots are provided by L₁ =log(y₁) and L₂=log(y₂) and the corresponding error values are provided by ##EQU10##and E₂ =S₀ +E₁. The steps of the process of FIG. 21 are performed untilall error values and locations of all interleaves are computed. Theerror values and error locations are stored in holding registers 105.

After the error values and locations are computed, burstlimiting anderror corrections are performed. Burstlimiting and error corrections areperformed by burstlimiter 103. Burstlimiter 103 includes a binaryarithmetic unit, which is controlled by a state machine provided by aROM and an instruction decode unit. The state machine of burstlimiter103 operates under the processes described below in conjunction with theflow diagrams of FIGS. 22, 23a and 23b. The state machine and the binaryarithmetic unit in burstlimiter 103 are provided by random logiccircuits.

Error locations computed in accordance of the process of FIG. 21 areprovided as locations within each interleave. FIG. 22 is a flow diagramshowing a process for (a) converting error locations within eachinterleave to error locations relative to the beginning of the checksymbol field in a 1024-byte sector; and (b) sorting the error locationsin ascending order. The process of FIG. 2 prepares the data for theburstlimiting functions of burstlimiter 103. As shown in FIG. 22, toprovide such error locations with respect to the 1024-byte sector, asenumerated from the beginning of the check symbol field, each errorlocation is multiplied by three and an appropriate offset (i.e. 0, 1 or2) is added to each error location, according to the interleave in whichthe error is located (step 2202). (See FIG. 24 regarding the positionenumeration scheme used in this description). The error locations arethen sorted in ascending order for the application of the burstlimitingand error correction procedures discussed below with respect to FIGS.23a and 23b. The error locations can be sorted by any sorting algorithm.FIG. 23 shows, for example, as indicated generally by reference numerals2203, a conventional "bubble sort" algorithm.

FIGS. 23a and 23b together form a flow diagram showing the burstlimitingand error correction operations of burstlimiter 103. As shown in FIG.23a, at step 20, the variables BURSTNUM, NEXTBYTE and NEXTBIT, STOREREGand I are initialized to zero. BURSTNUM is a running count of the numberof burst errors found. NEXTBYTE is a running pointer to the next errorlocation. NEXTBIT is a running pointer pointing to a bit within an 8-bitdata symbol. The combination of NEXTBYTE and NEXTBIT identifies the bitin the 1024-byte sector currently being examined. STOREREG is atemporary variable for error location computation. I is a index pointerpointing to one of the six error location registers in holding registers105. Prior to step 1 of the process shown in FIG. 23a, the errorlocations have been provided in ascending order of index pointer I bythe sorting step 2203 of FIG. 22. In the description below, bytepositions and bit positions are numbered in the manner shown in FIG. 24.

Returning to FIG. 23a, at step 21, the value in the next error location(i.e. the value in the error location register in holding registers 105,which is pointed to by index pointer I), is compared to the value 523.In this embodiment, in a 1024-byte sector, bytes 0 to 523 are the dataand check symbols of the Reed-Solomon code word, and bytes 524 to 768are provided zeroes. If the next error location is greater than 523,then step 23 examines if this next error location is greater than 768.If this next error location is greater than 768, which is a preset valueto signal that the current error location register does not contain anerror location, the error correction procedure, beginning at step 2 ofFIG. 23b, can begin (step 33). Otherwise, the next error location, whichis a value between 523 and 768, indicates an uncorrectable error hasoccurred. An alternative error handling procedure, such as re-readingthe 1024-byte sector, can then be initiated by the disk-drivecontroller, which incorporates integrated circuit 100 of the presentembodiment, to remedy the uncorrectable error. Step 40 indicates a jumpto step 50 (FIG. 23b), which is a termination of the presentburstlimiting and error correction procedure.

If the next error location is less than 523, as determined by step 21,the next error location is compared to the NEXTBYTE variable (steps 22and 23). If the next error location is less than NEXTBYTE (i.e. thecurrent byte), a condition which is explained below, index pointer I isincremented so as to set the next error location to the value in thenext higher error location register. The variable BURSTNUM is thenchecked if the maximum allowable number of burst errors, which is two inthe present embodiment, is exceeded. An uncorrectable error is deemed tohave occurred if BURSTNUM exceeds two. Otherwise, if the index pointer Iequals 6, indicating that all error locations have been examined, theprocess jumps to step 2 of FIG. 23b. Step 2 is the beginning of theerror correction process.

At step 23, if the NEXTBYTE variable is at the error location, theprocess jumps to step 23. If the NEXTBYTE variable is less than the nexterror location, the variable NEXTBIT is reset to zero (step 30) and theprocess jumps (step 31) to step 23. At step 23, the variable NEXTBYTE isset to the next error location, and the STOREREG variable receives theerror value at the next error location right-shifted by the value of thevariable NEXTBIT. At step 25, the least significant bit (LSB) of thevariable STOREREG is tested. If this LSB is set, representing an errorat the NEXTBIT position of NEXTBYTE, a burst error is deemed to haveoccurred. At step 27, the variable BURSTNUM is incremented, and thevariables NEXTBYTE and NEXTBIT are set to the position offset from thecurrent position by the programmable single burst length. In the presentinvention, the single burst length is a user-programmable value from 1to 17. If another error occurs within this burst length, the conditionthat NEXTBYTE is greater than the error location pointed to by the nexthigher value of index pointer I is created. From step 27, the processjumps to step 38 to increment index pointer I.

At step 25 above, if the LSB of STOREREG is not set, i.e. the currentbit is not an error bit, the next error bit is searched by incrementingthe variable NEXTBIT (step 35) and right-shifting the variable STOREREG.If NEXTBIT is incremented to zero, the variable NEXTBYTE is incremented,so that (NEXTBYTE, NEXTBIT) points to the most significant bit at thenew value of NEXTBYTE. The process then branches to step 21 to allowexamination of the next error location. If NEXTBIT is not zero, thesearch for an error bit at NEXTBYTE is continued by the processreturning to step 25.

After all error locations are examined, i.e. at step 2, the errorcorrection process begins. At this time, the data field of theReed-Solomon code word is read into a buffer, with a buffer pointerpointing to the beginning of the data field. From step 2, the processproceeds to step 42 where the index pointer I is zeroed and variableSTOREREG is initialized to position 12 to point to the beginning of thedata field (see FIG. 24). If the next error location is found to bebetween 12 and 523 (steps 43 and 44), i.e. within the data field of thecurrent Reed-Solomon code word, the buffer pointer is reset to point tothe next error location (steps 46, 47 and 58). If the next errorlocation is found to be less than 12, the error is ignored and the indexpointer I is incremented (step 54) to point to the next error location,unless index pointer I equals 6 (step 55), which indicates that allerrors in the data field of the current Reed-Solomon code word have beencorrected. The process then terminates (step 56) until the next1024-sector is accessed. Likewise, when the next error location is foundto be outside of the data field (step 45), the process of FIG. 23bterminate.

When the buffer pointer is set to the next error location (i.e. aftercompleting steps 46, 47 and 58), the error value corresponding to theerror location is exclusive-ORed into the value pointed to by the bufferpointer to correct the error at the error location (steps 48, 57). Inthis embodiment, a control signal CREQ is asserted to initiate an errorcorrection (step 57). Acknowledgement signal CACK is asserted when thecorrection is completed (step 48). The process of FIG. 23b terminateswhen the value at the last error location is corrected (step 50).Otherwise, the variable STOREREG is set to the next error location,index pointer I is incremented (step 52) and the process returns to step44 (step 53).

The above detailed description is provided to illustrate the specificembodiments of the present invention and is not intended to be limiting.Numerous variations and modifications within the scope of the presentinvention are possible. The present invention is defined by the claimsset forth below.

We claim:
 1. An arithmetic and logic unit for performing logarithmicoperations over a GF(q⁴) finite Galois field using logic circuits forlogarithmic operations over a GF(q²) finite Galois field, wherein q canbe expressed as 2^(n), n≧1, n being an integer which is a power of two,and GF(q⁴) can be expressed as GF(q⁴)={x|x=aγ+b;a,bεGF(q²)}, said logiccircuits for logarithmic operations over a GF(q²) finite Galois fieldincluding logic circuits for logarithmic operations over a GF(q) finiteGalois field, wherein GF(q²) can be expressed asGF(q²)={x|x=cβ+d;c,dεGF(q)}, where γ and β are basis elements of GF(q⁴)and GF(q²) respectively, a and b being referred to as a first elementand a second element of an element in said GF(q⁴) finite Galois field,and said c and d being referred to as a first element and a secondelement of an element in said GF(q²) finite Galois field.
 2. A circuitas in claim 1, wherein q⁴ equals 256, such that said GF(q²) finiteGalois field can be represented by GF'(16)={x|x=aγ+b;a,bεGF'(4)}, whereδ is a basis element of said GF(256) finite Galois field in which δ²+δ+ε=0, and ε=βγ+γ, for some γ and β belonging to a GF(16) finite Galoisfield and a GF(4) finite Galois field respectively.
 3. An arithmetic andlogic unit for performing multiplicative inverse operations over aGF(q⁴) finite Galois field using logic circuits for multiplicativeinverse operations over a GF(q²) finite Galois field, wherein q, can beexpressed as 2^(n), n≧1, n being an integer which is a power of two, andGF(q⁴) can be expressed as GF(q⁴)={x|x=aγ+b;a,bεGF(q²)}, said logiccircuits for multiplicative inverse operations over a GF(q²) finiteGalois field including logic circuits for multiplicative inverseoperations over a GF(q) finite Galois field, wherein GF(q²) can beexpressed as GF(q²)={x|x=cβ+d;c,dεGF(q)}, where γ and β are basiselements of GF(q⁴) and GF(q²) respectively, a and b being referred to asa first element and a second element of an element in said GF(q⁴) finiteGalois field, and said c and d being referred to as a first element anda second element of an element in said GF(q²) finite Galois field.
 4. Acircuit as in claim 3, wherein q⁴ equals 256, such that said GF(q²)finite Galois field can be represented by GF'(16)={x|x=aγ+b;a,bεGF'(4)},where δ is a basis element of said GF(256) finite Galois field in whichδ² +δ+ε=0, and ε=βγ+γ, for some γ and β belonging to a GF(16) finiteGalois field and a GF(4) finite Galois field respectively.